Copied to
clipboard

## G = Dic5.13D12order 480 = 25·3·5

### 9th non-split extension by Dic5 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic5.13D12
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C2×C15⋊C8 — Dic5.13D12
 Lower central C15 — C30 — Dic5.13D12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic5.13D12
G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5b, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=a5c-1 >

Subgroups: 284 in 76 conjugacy classes, 41 normal (35 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C4⋊C8, C5⋊C8, C2×Dic5, C2×C20, C2×C3⋊C8, C4×C12, C3×Dic5, C3×Dic5, C60, C2×C30, C4×Dic5, C2×C5⋊C8, C12⋊C8, C15⋊C8, C6×Dic5, C2×C60, Dic5⋊C8, C12×Dic5, C2×C15⋊C8, Dic5.13D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C2×C8, M4(2), F5, C3⋊C8, Dic6, D12, C2×Dic3, C4⋊C8, C2×F5, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C3⋊F5, D5⋊C8, C4⋊F5, C22.F5, C12⋊C8, C2×C3⋊F5, Dic5⋊C8, C60.C4, C60⋊C4, C158M4(2), Dic5.13D12

Smallest permutation representation of Dic5.13D12
Regular action on 480 points
Generators in S480
(1 443 388 380 319 231 468 426 454 251)(2 444 389 381 320 232 457 427 455 252)(3 433 390 382 321 233 458 428 456 241)(4 434 391 383 322 234 459 429 445 242)(5 435 392 384 323 235 460 430 446 243)(6 436 393 373 324 236 461 431 447 244)(7 437 394 374 313 237 462 432 448 245)(8 438 395 375 314 238 463 421 449 246)(9 439 396 376 315 239 464 422 450 247)(10 440 385 377 316 240 465 423 451 248)(11 441 386 378 317 229 466 424 452 249)(12 442 387 379 318 230 467 425 453 250)(13 274 52 117 101 88 69 32 73 196)(14 275 53 118 102 89 70 33 74 197)(15 276 54 119 103 90 71 34 75 198)(16 265 55 120 104 91 72 35 76 199)(17 266 56 109 105 92 61 36 77 200)(18 267 57 110 106 93 62 25 78 201)(19 268 58 111 107 94 63 26 79 202)(20 269 59 112 108 95 64 27 80 203)(21 270 60 113 97 96 65 28 81 204)(22 271 49 114 98 85 66 29 82 193)(23 272 50 115 99 86 67 30 83 194)(24 273 51 116 100 87 68 31 84 195)(37 147 144 366 475 183 345 171 298 411)(38 148 133 367 476 184 346 172 299 412)(39 149 134 368 477 185 347 173 300 413)(40 150 135 369 478 186 348 174 289 414)(41 151 136 370 479 187 337 175 290 415)(42 152 137 371 480 188 338 176 291 416)(43 153 138 372 469 189 339 177 292 417)(44 154 139 361 470 190 340 178 293 418)(45 155 140 362 471 191 341 179 294 419)(46 156 141 363 472 192 342 180 295 420)(47 145 142 364 473 181 343 169 296 409)(48 146 143 365 474 182 344 170 297 410)(121 163 256 208 352 401 333 311 281 227)(122 164 257 209 353 402 334 312 282 228)(123 165 258 210 354 403 335 301 283 217)(124 166 259 211 355 404 336 302 284 218)(125 167 260 212 356 405 325 303 285 219)(126 168 261 213 357 406 326 304 286 220)(127 157 262 214 358 407 327 305 287 221)(128 158 263 215 359 408 328 306 288 222)(129 159 264 216 360 397 329 307 277 223)(130 160 253 205 349 398 330 308 278 224)(131 161 254 206 350 399 331 309 279 225)(132 162 255 207 351 400 332 310 280 226)
(1 325 231 167)(2 326 232 168)(3 327 233 157)(4 328 234 158)(5 329 235 159)(6 330 236 160)(7 331 237 161)(8 332 238 162)(9 333 239 163)(10 334 240 164)(11 335 229 165)(12 336 230 166)(13 469 88 417)(14 470 89 418)(15 471 90 419)(16 472 91 420)(17 473 92 409)(18 474 93 410)(19 475 94 411)(20 476 95 412)(21 477 96 413)(22 478 85 414)(23 479 86 415)(24 480 87 416)(25 170 57 143)(26 171 58 144)(27 172 59 133)(28 173 60 134)(29 174 49 135)(30 175 50 136)(31 176 51 137)(32 177 52 138)(33 178 53 139)(34 179 54 140)(35 180 55 141)(36 169 56 142)(37 202 183 107)(38 203 184 108)(39 204 185 97)(40 193 186 98)(41 194 187 99)(42 195 188 100)(43 196 189 101)(44 197 190 102)(45 198 191 103)(46 199 192 104)(47 200 181 105)(48 201 182 106)(61 296 266 364)(62 297 267 365)(63 298 268 366)(64 299 269 367)(65 300 270 368)(66 289 271 369)(67 290 272 370)(68 291 273 371)(69 292 274 372)(70 293 275 361)(71 294 276 362)(72 295 265 363)(73 339 117 153)(74 340 118 154)(75 341 119 155)(76 342 120 156)(77 343 109 145)(78 344 110 146)(79 345 111 147)(80 346 112 148)(81 347 113 149)(82 348 114 150)(83 337 115 151)(84 338 116 152)(121 439 401 464)(122 440 402 465)(123 441 403 466)(124 442 404 467)(125 443 405 468)(126 444 406 457)(127 433 407 458)(128 434 408 459)(129 435 397 460)(130 436 398 461)(131 437 399 462)(132 438 400 463)(205 447 278 373)(206 448 279 374)(207 449 280 375)(208 450 281 376)(209 451 282 377)(210 452 283 378)(211 453 284 379)(212 454 285 380)(213 455 286 381)(214 456 287 382)(215 445 288 383)(216 446 277 384)(217 386 354 424)(218 387 355 425)(219 388 356 426)(220 389 357 427)(221 390 358 428)(222 391 359 429)(223 392 360 430)(224 393 349 431)(225 394 350 432)(226 395 351 421)(227 396 352 422)(228 385 353 423)(241 305 321 262)(242 306 322 263)(243 307 323 264)(244 308 324 253)(245 309 313 254)(246 310 314 255)(247 311 315 256)(248 312 316 257)(249 301 317 258)(250 302 318 259)(251 303 319 260)(252 304 320 261)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200 201 202 203 204)(205 206 207 208 209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252)(253 254 255 256 257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272 273 274 275 276)(277 278 279 280 281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320 321 322 323 324)(325 326 327 328 329 330 331 332 333 334 335 336)(337 338 339 340 341 342 343 344 345 346 347 348)(349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372)(373 374 375 376 377 378 379 380 381 382 383 384)(385 386 387 388 389 390 391 392 393 394 395 396)(397 398 399 400 401 402 403 404 405 406 407 408)(409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440 441 442 443 444)(445 446 447 448 449 450 451 452 453 454 455 456)(457 458 459 460 461 462 463 464 465 466 467 468)(469 470 471 472 473 474 475 476 477 478 479 480)
(1 269 167 299 231 64 325 367)(2 63 168 366 232 268 326 298)(3 267 157 297 233 62 327 365)(4 61 158 364 234 266 328 296)(5 265 159 295 235 72 329 363)(6 71 160 362 236 276 330 294)(7 275 161 293 237 70 331 361)(8 69 162 372 238 274 332 292)(9 273 163 291 239 68 333 371)(10 67 164 370 240 272 334 290)(11 271 165 289 229 66 335 369)(12 65 166 368 230 270 336 300)(13 207 417 375 88 280 469 449)(14 279 418 448 89 206 470 374)(15 205 419 373 90 278 471 447)(16 277 420 446 91 216 472 384)(17 215 409 383 92 288 473 445)(18 287 410 456 93 214 474 382)(19 213 411 381 94 286 475 455)(20 285 412 454 95 212 476 380)(21 211 413 379 96 284 477 453)(22 283 414 452 85 210 478 378)(23 209 415 377 86 282 479 451)(24 281 416 450 87 208 480 376)(25 221 143 428 57 358 170 390)(26 357 144 389 58 220 171 427)(27 219 133 426 59 356 172 388)(28 355 134 387 60 218 173 425)(29 217 135 424 49 354 174 386)(30 353 136 385 50 228 175 423)(31 227 137 422 51 352 176 396)(32 351 138 395 52 226 177 421)(33 225 139 432 53 350 178 394)(34 349 140 393 54 224 179 431)(35 223 141 430 55 360 180 392)(36 359 142 391 56 222 169 429)(37 457 107 406 183 444 202 126)(38 443 108 125 184 468 203 405)(39 467 97 404 185 442 204 124)(40 441 98 123 186 466 193 403)(41 465 99 402 187 440 194 122)(42 439 100 121 188 464 195 401)(43 463 101 400 189 438 196 132)(44 437 102 131 190 462 197 399)(45 461 103 398 191 436 198 130)(46 435 104 129 192 460 199 397)(47 459 105 408 181 434 200 128)(48 433 106 127 182 458 201 407)(73 310 153 246 117 255 339 314)(74 254 154 313 118 309 340 245)(75 308 155 244 119 253 341 324)(76 264 156 323 120 307 342 243)(77 306 145 242 109 263 343 322)(78 262 146 321 110 305 344 241)(79 304 147 252 111 261 345 320)(80 260 148 319 112 303 346 251)(81 302 149 250 113 259 347 318)(82 258 150 317 114 301 348 249)(83 312 151 248 115 257 337 316)(84 256 152 315 116 311 338 247)

G:=sub<Sym(480)| (1,443,388,380,319,231,468,426,454,251)(2,444,389,381,320,232,457,427,455,252)(3,433,390,382,321,233,458,428,456,241)(4,434,391,383,322,234,459,429,445,242)(5,435,392,384,323,235,460,430,446,243)(6,436,393,373,324,236,461,431,447,244)(7,437,394,374,313,237,462,432,448,245)(8,438,395,375,314,238,463,421,449,246)(9,439,396,376,315,239,464,422,450,247)(10,440,385,377,316,240,465,423,451,248)(11,441,386,378,317,229,466,424,452,249)(12,442,387,379,318,230,467,425,453,250)(13,274,52,117,101,88,69,32,73,196)(14,275,53,118,102,89,70,33,74,197)(15,276,54,119,103,90,71,34,75,198)(16,265,55,120,104,91,72,35,76,199)(17,266,56,109,105,92,61,36,77,200)(18,267,57,110,106,93,62,25,78,201)(19,268,58,111,107,94,63,26,79,202)(20,269,59,112,108,95,64,27,80,203)(21,270,60,113,97,96,65,28,81,204)(22,271,49,114,98,85,66,29,82,193)(23,272,50,115,99,86,67,30,83,194)(24,273,51,116,100,87,68,31,84,195)(37,147,144,366,475,183,345,171,298,411)(38,148,133,367,476,184,346,172,299,412)(39,149,134,368,477,185,347,173,300,413)(40,150,135,369,478,186,348,174,289,414)(41,151,136,370,479,187,337,175,290,415)(42,152,137,371,480,188,338,176,291,416)(43,153,138,372,469,189,339,177,292,417)(44,154,139,361,470,190,340,178,293,418)(45,155,140,362,471,191,341,179,294,419)(46,156,141,363,472,192,342,180,295,420)(47,145,142,364,473,181,343,169,296,409)(48,146,143,365,474,182,344,170,297,410)(121,163,256,208,352,401,333,311,281,227)(122,164,257,209,353,402,334,312,282,228)(123,165,258,210,354,403,335,301,283,217)(124,166,259,211,355,404,336,302,284,218)(125,167,260,212,356,405,325,303,285,219)(126,168,261,213,357,406,326,304,286,220)(127,157,262,214,358,407,327,305,287,221)(128,158,263,215,359,408,328,306,288,222)(129,159,264,216,360,397,329,307,277,223)(130,160,253,205,349,398,330,308,278,224)(131,161,254,206,350,399,331,309,279,225)(132,162,255,207,351,400,332,310,280,226), (1,325,231,167)(2,326,232,168)(3,327,233,157)(4,328,234,158)(5,329,235,159)(6,330,236,160)(7,331,237,161)(8,332,238,162)(9,333,239,163)(10,334,240,164)(11,335,229,165)(12,336,230,166)(13,469,88,417)(14,470,89,418)(15,471,90,419)(16,472,91,420)(17,473,92,409)(18,474,93,410)(19,475,94,411)(20,476,95,412)(21,477,96,413)(22,478,85,414)(23,479,86,415)(24,480,87,416)(25,170,57,143)(26,171,58,144)(27,172,59,133)(28,173,60,134)(29,174,49,135)(30,175,50,136)(31,176,51,137)(32,177,52,138)(33,178,53,139)(34,179,54,140)(35,180,55,141)(36,169,56,142)(37,202,183,107)(38,203,184,108)(39,204,185,97)(40,193,186,98)(41,194,187,99)(42,195,188,100)(43,196,189,101)(44,197,190,102)(45,198,191,103)(46,199,192,104)(47,200,181,105)(48,201,182,106)(61,296,266,364)(62,297,267,365)(63,298,268,366)(64,299,269,367)(65,300,270,368)(66,289,271,369)(67,290,272,370)(68,291,273,371)(69,292,274,372)(70,293,275,361)(71,294,276,362)(72,295,265,363)(73,339,117,153)(74,340,118,154)(75,341,119,155)(76,342,120,156)(77,343,109,145)(78,344,110,146)(79,345,111,147)(80,346,112,148)(81,347,113,149)(82,348,114,150)(83,337,115,151)(84,338,116,152)(121,439,401,464)(122,440,402,465)(123,441,403,466)(124,442,404,467)(125,443,405,468)(126,444,406,457)(127,433,407,458)(128,434,408,459)(129,435,397,460)(130,436,398,461)(131,437,399,462)(132,438,400,463)(205,447,278,373)(206,448,279,374)(207,449,280,375)(208,450,281,376)(209,451,282,377)(210,452,283,378)(211,453,284,379)(212,454,285,380)(213,455,286,381)(214,456,287,382)(215,445,288,383)(216,446,277,384)(217,386,354,424)(218,387,355,425)(219,388,356,426)(220,389,357,427)(221,390,358,428)(222,391,359,429)(223,392,360,430)(224,393,349,431)(225,394,350,432)(226,395,351,421)(227,396,352,422)(228,385,353,423)(241,305,321,262)(242,306,322,263)(243,307,323,264)(244,308,324,253)(245,309,313,254)(246,310,314,255)(247,311,315,256)(248,312,316,257)(249,301,317,258)(250,302,318,259)(251,303,319,260)(252,304,320,261), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272,273,274,275,276)(277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348)(349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372)(373,374,375,376,377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441,442,443,444)(445,446,447,448,449,450,451,452,453,454,455,456)(457,458,459,460,461,462,463,464,465,466,467,468)(469,470,471,472,473,474,475,476,477,478,479,480), (1,269,167,299,231,64,325,367)(2,63,168,366,232,268,326,298)(3,267,157,297,233,62,327,365)(4,61,158,364,234,266,328,296)(5,265,159,295,235,72,329,363)(6,71,160,362,236,276,330,294)(7,275,161,293,237,70,331,361)(8,69,162,372,238,274,332,292)(9,273,163,291,239,68,333,371)(10,67,164,370,240,272,334,290)(11,271,165,289,229,66,335,369)(12,65,166,368,230,270,336,300)(13,207,417,375,88,280,469,449)(14,279,418,448,89,206,470,374)(15,205,419,373,90,278,471,447)(16,277,420,446,91,216,472,384)(17,215,409,383,92,288,473,445)(18,287,410,456,93,214,474,382)(19,213,411,381,94,286,475,455)(20,285,412,454,95,212,476,380)(21,211,413,379,96,284,477,453)(22,283,414,452,85,210,478,378)(23,209,415,377,86,282,479,451)(24,281,416,450,87,208,480,376)(25,221,143,428,57,358,170,390)(26,357,144,389,58,220,171,427)(27,219,133,426,59,356,172,388)(28,355,134,387,60,218,173,425)(29,217,135,424,49,354,174,386)(30,353,136,385,50,228,175,423)(31,227,137,422,51,352,176,396)(32,351,138,395,52,226,177,421)(33,225,139,432,53,350,178,394)(34,349,140,393,54,224,179,431)(35,223,141,430,55,360,180,392)(36,359,142,391,56,222,169,429)(37,457,107,406,183,444,202,126)(38,443,108,125,184,468,203,405)(39,467,97,404,185,442,204,124)(40,441,98,123,186,466,193,403)(41,465,99,402,187,440,194,122)(42,439,100,121,188,464,195,401)(43,463,101,400,189,438,196,132)(44,437,102,131,190,462,197,399)(45,461,103,398,191,436,198,130)(46,435,104,129,192,460,199,397)(47,459,105,408,181,434,200,128)(48,433,106,127,182,458,201,407)(73,310,153,246,117,255,339,314)(74,254,154,313,118,309,340,245)(75,308,155,244,119,253,341,324)(76,264,156,323,120,307,342,243)(77,306,145,242,109,263,343,322)(78,262,146,321,110,305,344,241)(79,304,147,252,111,261,345,320)(80,260,148,319,112,303,346,251)(81,302,149,250,113,259,347,318)(82,258,150,317,114,301,348,249)(83,312,151,248,115,257,337,316)(84,256,152,315,116,311,338,247)>;

G:=Group( (1,443,388,380,319,231,468,426,454,251)(2,444,389,381,320,232,457,427,455,252)(3,433,390,382,321,233,458,428,456,241)(4,434,391,383,322,234,459,429,445,242)(5,435,392,384,323,235,460,430,446,243)(6,436,393,373,324,236,461,431,447,244)(7,437,394,374,313,237,462,432,448,245)(8,438,395,375,314,238,463,421,449,246)(9,439,396,376,315,239,464,422,450,247)(10,440,385,377,316,240,465,423,451,248)(11,441,386,378,317,229,466,424,452,249)(12,442,387,379,318,230,467,425,453,250)(13,274,52,117,101,88,69,32,73,196)(14,275,53,118,102,89,70,33,74,197)(15,276,54,119,103,90,71,34,75,198)(16,265,55,120,104,91,72,35,76,199)(17,266,56,109,105,92,61,36,77,200)(18,267,57,110,106,93,62,25,78,201)(19,268,58,111,107,94,63,26,79,202)(20,269,59,112,108,95,64,27,80,203)(21,270,60,113,97,96,65,28,81,204)(22,271,49,114,98,85,66,29,82,193)(23,272,50,115,99,86,67,30,83,194)(24,273,51,116,100,87,68,31,84,195)(37,147,144,366,475,183,345,171,298,411)(38,148,133,367,476,184,346,172,299,412)(39,149,134,368,477,185,347,173,300,413)(40,150,135,369,478,186,348,174,289,414)(41,151,136,370,479,187,337,175,290,415)(42,152,137,371,480,188,338,176,291,416)(43,153,138,372,469,189,339,177,292,417)(44,154,139,361,470,190,340,178,293,418)(45,155,140,362,471,191,341,179,294,419)(46,156,141,363,472,192,342,180,295,420)(47,145,142,364,473,181,343,169,296,409)(48,146,143,365,474,182,344,170,297,410)(121,163,256,208,352,401,333,311,281,227)(122,164,257,209,353,402,334,312,282,228)(123,165,258,210,354,403,335,301,283,217)(124,166,259,211,355,404,336,302,284,218)(125,167,260,212,356,405,325,303,285,219)(126,168,261,213,357,406,326,304,286,220)(127,157,262,214,358,407,327,305,287,221)(128,158,263,215,359,408,328,306,288,222)(129,159,264,216,360,397,329,307,277,223)(130,160,253,205,349,398,330,308,278,224)(131,161,254,206,350,399,331,309,279,225)(132,162,255,207,351,400,332,310,280,226), (1,325,231,167)(2,326,232,168)(3,327,233,157)(4,328,234,158)(5,329,235,159)(6,330,236,160)(7,331,237,161)(8,332,238,162)(9,333,239,163)(10,334,240,164)(11,335,229,165)(12,336,230,166)(13,469,88,417)(14,470,89,418)(15,471,90,419)(16,472,91,420)(17,473,92,409)(18,474,93,410)(19,475,94,411)(20,476,95,412)(21,477,96,413)(22,478,85,414)(23,479,86,415)(24,480,87,416)(25,170,57,143)(26,171,58,144)(27,172,59,133)(28,173,60,134)(29,174,49,135)(30,175,50,136)(31,176,51,137)(32,177,52,138)(33,178,53,139)(34,179,54,140)(35,180,55,141)(36,169,56,142)(37,202,183,107)(38,203,184,108)(39,204,185,97)(40,193,186,98)(41,194,187,99)(42,195,188,100)(43,196,189,101)(44,197,190,102)(45,198,191,103)(46,199,192,104)(47,200,181,105)(48,201,182,106)(61,296,266,364)(62,297,267,365)(63,298,268,366)(64,299,269,367)(65,300,270,368)(66,289,271,369)(67,290,272,370)(68,291,273,371)(69,292,274,372)(70,293,275,361)(71,294,276,362)(72,295,265,363)(73,339,117,153)(74,340,118,154)(75,341,119,155)(76,342,120,156)(77,343,109,145)(78,344,110,146)(79,345,111,147)(80,346,112,148)(81,347,113,149)(82,348,114,150)(83,337,115,151)(84,338,116,152)(121,439,401,464)(122,440,402,465)(123,441,403,466)(124,442,404,467)(125,443,405,468)(126,444,406,457)(127,433,407,458)(128,434,408,459)(129,435,397,460)(130,436,398,461)(131,437,399,462)(132,438,400,463)(205,447,278,373)(206,448,279,374)(207,449,280,375)(208,450,281,376)(209,451,282,377)(210,452,283,378)(211,453,284,379)(212,454,285,380)(213,455,286,381)(214,456,287,382)(215,445,288,383)(216,446,277,384)(217,386,354,424)(218,387,355,425)(219,388,356,426)(220,389,357,427)(221,390,358,428)(222,391,359,429)(223,392,360,430)(224,393,349,431)(225,394,350,432)(226,395,351,421)(227,396,352,422)(228,385,353,423)(241,305,321,262)(242,306,322,263)(243,307,323,264)(244,308,324,253)(245,309,313,254)(246,310,314,255)(247,311,315,256)(248,312,316,257)(249,301,317,258)(250,302,318,259)(251,303,319,260)(252,304,320,261), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250,251,252)(253,254,255,256,257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272,273,274,275,276)(277,278,279,280,281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296,297,298,299,300)(301,302,303,304,305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320,321,322,323,324)(325,326,327,328,329,330,331,332,333,334,335,336)(337,338,339,340,341,342,343,344,345,346,347,348)(349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372)(373,374,375,376,377,378,379,380,381,382,383,384)(385,386,387,388,389,390,391,392,393,394,395,396)(397,398,399,400,401,402,403,404,405,406,407,408)(409,410,411,412,413,414,415,416,417,418,419,420)(421,422,423,424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441,442,443,444)(445,446,447,448,449,450,451,452,453,454,455,456)(457,458,459,460,461,462,463,464,465,466,467,468)(469,470,471,472,473,474,475,476,477,478,479,480), (1,269,167,299,231,64,325,367)(2,63,168,366,232,268,326,298)(3,267,157,297,233,62,327,365)(4,61,158,364,234,266,328,296)(5,265,159,295,235,72,329,363)(6,71,160,362,236,276,330,294)(7,275,161,293,237,70,331,361)(8,69,162,372,238,274,332,292)(9,273,163,291,239,68,333,371)(10,67,164,370,240,272,334,290)(11,271,165,289,229,66,335,369)(12,65,166,368,230,270,336,300)(13,207,417,375,88,280,469,449)(14,279,418,448,89,206,470,374)(15,205,419,373,90,278,471,447)(16,277,420,446,91,216,472,384)(17,215,409,383,92,288,473,445)(18,287,410,456,93,214,474,382)(19,213,411,381,94,286,475,455)(20,285,412,454,95,212,476,380)(21,211,413,379,96,284,477,453)(22,283,414,452,85,210,478,378)(23,209,415,377,86,282,479,451)(24,281,416,450,87,208,480,376)(25,221,143,428,57,358,170,390)(26,357,144,389,58,220,171,427)(27,219,133,426,59,356,172,388)(28,355,134,387,60,218,173,425)(29,217,135,424,49,354,174,386)(30,353,136,385,50,228,175,423)(31,227,137,422,51,352,176,396)(32,351,138,395,52,226,177,421)(33,225,139,432,53,350,178,394)(34,349,140,393,54,224,179,431)(35,223,141,430,55,360,180,392)(36,359,142,391,56,222,169,429)(37,457,107,406,183,444,202,126)(38,443,108,125,184,468,203,405)(39,467,97,404,185,442,204,124)(40,441,98,123,186,466,193,403)(41,465,99,402,187,440,194,122)(42,439,100,121,188,464,195,401)(43,463,101,400,189,438,196,132)(44,437,102,131,190,462,197,399)(45,461,103,398,191,436,198,130)(46,435,104,129,192,460,199,397)(47,459,105,408,181,434,200,128)(48,433,106,127,182,458,201,407)(73,310,153,246,117,255,339,314)(74,254,154,313,118,309,340,245)(75,308,155,244,119,253,341,324)(76,264,156,323,120,307,342,243)(77,306,145,242,109,263,343,322)(78,262,146,321,110,305,344,241)(79,304,147,252,111,261,345,320)(80,260,148,319,112,303,346,251)(81,302,149,250,113,259,347,318)(82,258,150,317,114,301,348,249)(83,312,151,248,115,257,337,316)(84,256,152,315,116,311,338,247) );

G=PermutationGroup([[(1,443,388,380,319,231,468,426,454,251),(2,444,389,381,320,232,457,427,455,252),(3,433,390,382,321,233,458,428,456,241),(4,434,391,383,322,234,459,429,445,242),(5,435,392,384,323,235,460,430,446,243),(6,436,393,373,324,236,461,431,447,244),(7,437,394,374,313,237,462,432,448,245),(8,438,395,375,314,238,463,421,449,246),(9,439,396,376,315,239,464,422,450,247),(10,440,385,377,316,240,465,423,451,248),(11,441,386,378,317,229,466,424,452,249),(12,442,387,379,318,230,467,425,453,250),(13,274,52,117,101,88,69,32,73,196),(14,275,53,118,102,89,70,33,74,197),(15,276,54,119,103,90,71,34,75,198),(16,265,55,120,104,91,72,35,76,199),(17,266,56,109,105,92,61,36,77,200),(18,267,57,110,106,93,62,25,78,201),(19,268,58,111,107,94,63,26,79,202),(20,269,59,112,108,95,64,27,80,203),(21,270,60,113,97,96,65,28,81,204),(22,271,49,114,98,85,66,29,82,193),(23,272,50,115,99,86,67,30,83,194),(24,273,51,116,100,87,68,31,84,195),(37,147,144,366,475,183,345,171,298,411),(38,148,133,367,476,184,346,172,299,412),(39,149,134,368,477,185,347,173,300,413),(40,150,135,369,478,186,348,174,289,414),(41,151,136,370,479,187,337,175,290,415),(42,152,137,371,480,188,338,176,291,416),(43,153,138,372,469,189,339,177,292,417),(44,154,139,361,470,190,340,178,293,418),(45,155,140,362,471,191,341,179,294,419),(46,156,141,363,472,192,342,180,295,420),(47,145,142,364,473,181,343,169,296,409),(48,146,143,365,474,182,344,170,297,410),(121,163,256,208,352,401,333,311,281,227),(122,164,257,209,353,402,334,312,282,228),(123,165,258,210,354,403,335,301,283,217),(124,166,259,211,355,404,336,302,284,218),(125,167,260,212,356,405,325,303,285,219),(126,168,261,213,357,406,326,304,286,220),(127,157,262,214,358,407,327,305,287,221),(128,158,263,215,359,408,328,306,288,222),(129,159,264,216,360,397,329,307,277,223),(130,160,253,205,349,398,330,308,278,224),(131,161,254,206,350,399,331,309,279,225),(132,162,255,207,351,400,332,310,280,226)], [(1,325,231,167),(2,326,232,168),(3,327,233,157),(4,328,234,158),(5,329,235,159),(6,330,236,160),(7,331,237,161),(8,332,238,162),(9,333,239,163),(10,334,240,164),(11,335,229,165),(12,336,230,166),(13,469,88,417),(14,470,89,418),(15,471,90,419),(16,472,91,420),(17,473,92,409),(18,474,93,410),(19,475,94,411),(20,476,95,412),(21,477,96,413),(22,478,85,414),(23,479,86,415),(24,480,87,416),(25,170,57,143),(26,171,58,144),(27,172,59,133),(28,173,60,134),(29,174,49,135),(30,175,50,136),(31,176,51,137),(32,177,52,138),(33,178,53,139),(34,179,54,140),(35,180,55,141),(36,169,56,142),(37,202,183,107),(38,203,184,108),(39,204,185,97),(40,193,186,98),(41,194,187,99),(42,195,188,100),(43,196,189,101),(44,197,190,102),(45,198,191,103),(46,199,192,104),(47,200,181,105),(48,201,182,106),(61,296,266,364),(62,297,267,365),(63,298,268,366),(64,299,269,367),(65,300,270,368),(66,289,271,369),(67,290,272,370),(68,291,273,371),(69,292,274,372),(70,293,275,361),(71,294,276,362),(72,295,265,363),(73,339,117,153),(74,340,118,154),(75,341,119,155),(76,342,120,156),(77,343,109,145),(78,344,110,146),(79,345,111,147),(80,346,112,148),(81,347,113,149),(82,348,114,150),(83,337,115,151),(84,338,116,152),(121,439,401,464),(122,440,402,465),(123,441,403,466),(124,442,404,467),(125,443,405,468),(126,444,406,457),(127,433,407,458),(128,434,408,459),(129,435,397,460),(130,436,398,461),(131,437,399,462),(132,438,400,463),(205,447,278,373),(206,448,279,374),(207,449,280,375),(208,450,281,376),(209,451,282,377),(210,452,283,378),(211,453,284,379),(212,454,285,380),(213,455,286,381),(214,456,287,382),(215,445,288,383),(216,446,277,384),(217,386,354,424),(218,387,355,425),(219,388,356,426),(220,389,357,427),(221,390,358,428),(222,391,359,429),(223,392,360,430),(224,393,349,431),(225,394,350,432),(226,395,351,421),(227,396,352,422),(228,385,353,423),(241,305,321,262),(242,306,322,263),(243,307,323,264),(244,308,324,253),(245,309,313,254),(246,310,314,255),(247,311,315,256),(248,312,316,257),(249,301,317,258),(250,302,318,259),(251,303,319,260),(252,304,320,261)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200,201,202,203,204),(205,206,207,208,209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250,251,252),(253,254,255,256,257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272,273,274,275,276),(277,278,279,280,281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296,297,298,299,300),(301,302,303,304,305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320,321,322,323,324),(325,326,327,328,329,330,331,332,333,334,335,336),(337,338,339,340,341,342,343,344,345,346,347,348),(349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372),(373,374,375,376,377,378,379,380,381,382,383,384),(385,386,387,388,389,390,391,392,393,394,395,396),(397,398,399,400,401,402,403,404,405,406,407,408),(409,410,411,412,413,414,415,416,417,418,419,420),(421,422,423,424,425,426,427,428,429,430,431,432),(433,434,435,436,437,438,439,440,441,442,443,444),(445,446,447,448,449,450,451,452,453,454,455,456),(457,458,459,460,461,462,463,464,465,466,467,468),(469,470,471,472,473,474,475,476,477,478,479,480)], [(1,269,167,299,231,64,325,367),(2,63,168,366,232,268,326,298),(3,267,157,297,233,62,327,365),(4,61,158,364,234,266,328,296),(5,265,159,295,235,72,329,363),(6,71,160,362,236,276,330,294),(7,275,161,293,237,70,331,361),(8,69,162,372,238,274,332,292),(9,273,163,291,239,68,333,371),(10,67,164,370,240,272,334,290),(11,271,165,289,229,66,335,369),(12,65,166,368,230,270,336,300),(13,207,417,375,88,280,469,449),(14,279,418,448,89,206,470,374),(15,205,419,373,90,278,471,447),(16,277,420,446,91,216,472,384),(17,215,409,383,92,288,473,445),(18,287,410,456,93,214,474,382),(19,213,411,381,94,286,475,455),(20,285,412,454,95,212,476,380),(21,211,413,379,96,284,477,453),(22,283,414,452,85,210,478,378),(23,209,415,377,86,282,479,451),(24,281,416,450,87,208,480,376),(25,221,143,428,57,358,170,390),(26,357,144,389,58,220,171,427),(27,219,133,426,59,356,172,388),(28,355,134,387,60,218,173,425),(29,217,135,424,49,354,174,386),(30,353,136,385,50,228,175,423),(31,227,137,422,51,352,176,396),(32,351,138,395,52,226,177,421),(33,225,139,432,53,350,178,394),(34,349,140,393,54,224,179,431),(35,223,141,430,55,360,180,392),(36,359,142,391,56,222,169,429),(37,457,107,406,183,444,202,126),(38,443,108,125,184,468,203,405),(39,467,97,404,185,442,204,124),(40,441,98,123,186,466,193,403),(41,465,99,402,187,440,194,122),(42,439,100,121,188,464,195,401),(43,463,101,400,189,438,196,132),(44,437,102,131,190,462,197,399),(45,461,103,398,191,436,198,130),(46,435,104,129,192,460,199,397),(47,459,105,408,181,434,200,128),(48,433,106,127,182,458,201,407),(73,310,153,246,117,255,339,314),(74,254,154,313,118,309,340,245),(75,308,155,244,119,253,341,324),(76,264,156,323,120,307,342,243),(77,306,145,242,109,263,343,322),(78,262,146,321,110,305,344,241),(79,304,147,252,111,261,345,320),(80,260,148,319,112,303,346,251),(81,302,149,250,113,259,347,318),(82,258,150,317,114,301,348,249),(83,312,151,248,115,257,337,316),(84,256,152,315,116,311,338,247)]])

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 8A ··· 8H 10A 10B 10C 12A 12B 12C 12D 12E ··· 12L 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 6 6 6 8 ··· 8 10 10 10 12 12 12 12 12 ··· 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 5 5 5 5 10 10 4 2 2 2 30 ··· 30 4 4 4 2 2 2 2 10 ··· 10 4 4 4 4 4 4 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - - + - - + + + - image C1 C2 C2 C4 C4 C8 S3 D4 Q8 Dic3 D6 Dic3 M4(2) C3⋊C8 Dic6 D12 C4.Dic3 F5 C2×F5 C3⋊F5 D5⋊C8 C4⋊F5 C22.F5 C2×C3⋊F5 C60.C4 C60⋊C4 C15⋊8M4(2) kernel Dic5.13D12 C12×Dic5 C2×C15⋊C8 C6×Dic5 C2×C60 C3×Dic5 C4×Dic5 C3×Dic5 C3×Dic5 C2×Dic5 C2×Dic5 C2×C20 C30 Dic5 Dic5 Dic5 C10 C2×C12 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 2 2 8 1 1 1 1 1 1 2 4 2 2 4 1 1 2 2 2 2 2 4 4 4

Matrix representation of Dic5.13D12 in GL8(𝔽241)

 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240
,
 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 176 132 80 0 0 0 0 164 120 68 229 0 0 0 0 197 145 65 77 0 0 0 0 189 109 121 44
,
 64 64 0 0 0 0 0 0 177 0 0 0 0 0 0 0 0 0 85 5 0 0 0 0 0 0 49 156 0 0 0 0 0 0 0 0 114 0 229 229 0 0 0 0 12 126 12 0 0 0 0 0 0 12 126 12 0 0 0 0 229 229 0 114
,
 123 227 0 0 0 0 0 0 104 118 0 0 0 0 0 0 0 0 200 51 0 0 0 0 0 0 123 41 0 0 0 0 0 0 0 0 17 66 91 65 0 0 0 0 215 150 167 216 0 0 0 0 49 74 48 224 0 0 0 0 176 193 1 26

G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240],[64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,164,197,189,0,0,0,0,176,120,145,109,0,0,0,0,132,68,65,121,0,0,0,0,80,229,77,44],[64,177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,85,49,0,0,0,0,0,0,5,156,0,0,0,0,0,0,0,0,114,12,0,229,0,0,0,0,0,126,12,229,0,0,0,0,229,12,126,0,0,0,0,0,229,0,12,114],[123,104,0,0,0,0,0,0,227,118,0,0,0,0,0,0,0,0,200,123,0,0,0,0,0,0,51,41,0,0,0,0,0,0,0,0,17,215,49,176,0,0,0,0,66,150,74,193,0,0,0,0,91,167,48,1,0,0,0,0,65,216,224,26] >;

Dic5.13D12 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{13}D_{12}
% in TeX

G:=Group("Dic5.13D12");
// GroupNames label

G:=SmallGroup(480,309);
// by ID

G=gap.SmallGroup(480,309);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,100,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=a^5*b,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

׿
×
𝔽